Metric properties
2010015810
Level:
B
The picture shows a square pyramid. The side of a base square is
\(a = 10\; \mathrm{cm}\) and the height of
the pyramid is \(v = 10\; \mathrm{cm}\).
Find the angle \(\varphi \) between the lateral edge and the edge of the base of the pyramid.
\(\mathop{\mathrm{tg}}\nolimits {\varphi} = \sqrt5
\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 65^{\circ }54^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{\sqrt5}
{5}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 24^{\circ }6^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}{2} = \frac{\sqrt5}
{5}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 48^{\circ }11^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits {\varphi} = \frac{\sqrt{10}}
{2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 57^{\circ }41^{\prime}\)
2010015809
Level:
B
The picture shows a square pyramid \(ABCDV\). The side of a base square is
\(a = 6\; \mathrm{cm}\) and the height of
the pyramid is \(v = 8\; \mathrm{cm}\).
Find the angle \(\varphi \) between the opposite lateral edges (the angle \(AVC\)).
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}2 = \frac{3\sqrt2}
{8}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 55^{\circ }53'\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{3\sqrt2}
{8}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 27^{\circ }56^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}{2} = \frac{3}
{8}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 41^{\circ }7^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}2 = \frac{8}
{3\sqrt2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 124^{\circ }7^{\prime}\)
2010015808
Level:
B
The picture shows a square pyramid. The side of a base square is
\(a = 6\; \mathrm{cm}\) and the height of
the pyramid is \(v = 10\; \mathrm{cm}\).
Find the angle \(\varphi \).
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{10}
{3\sqrt2}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 67^{\circ }\)
\(\mathop{\mathrm{tg}}\nolimits \varphi = \frac{10}
{3}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 73^{\circ }18^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}{2} = \frac{3\sqrt2}
{10}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 45^{\circ }59^{\prime}\)
\(\mathop{\mathrm{tg}}\nolimits \frac{\varphi}2 = \frac{3}
{10}\mathrel{\implies }\varphi \mathop{\mathop{\doteq }}\nolimits 33^{\circ }24^{\prime}\)
2010015807
Level:
A
The sides of a rectangular box shown in the picture are \(a = 3\, \mathrm{cm}\),
\(b = 4\, \mathrm{cm}\), and
\(c = 12\, \mathrm{cm}\). The space diagonal
is \(u_{t}\) and the shortest
face diagonal is \(u_{s}\).
Find the ratio \(u_{s} : u_{t}\).
\(5 : 13\)
\(13 : 5\)
\(13\sqrt{10}:40\)
\(4\sqrt{10}:13\)
2010015806
Level:
C
The side of a regular hexagonal prism
\(ABCDEFA'B'C'D'E'F'\) shown in the picture is
\(a = 3\, \mathrm{cm}\) and the height is
\(v = 8\, \mathrm{cm}\). Find the angle
between the diagonal \(AC'\)
and the base plane \(ABC\)
(round your result to the nearest degree).
\(57^{\circ }\)
\(53^{\circ }\)
\(33^{\circ }\)
\(38^{\circ }\)
2010015805
Level:
A
A cuboid has sides \(a = 6\, \mathrm{cm}\) and
\(b = 8\, \mathrm{cm}\), and the space diagonal
\(u = 11\, \mathrm{cm}\). Find the length of the side \(c\) (see the picture).
\( \sqrt{21}\,\mathrm{cm} \)
\( \sqrt{221}\,\mathrm{cm} \)
\( 21\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
2010015804
Level:
B
The base \( ABCD \) of a square pyramid \( ABCDV \) has an edge of \( 6\,\mathrm{cm} \). The height of the pyramid is \( 3\sqrt2\,\mathrm{cm} \). Find the distance between the point \( A \) and the line \( CV \) (see the picture).
\( 6\,\mathrm{cm} \)
\( 3\sqrt{3}\,\mathrm{cm} \)
\( 9\,\mathrm{cm} \)
\( 3\sqrt{2}\,\mathrm{cm} \)
2010015803
Level:
C
Let \( ABCD \) be a regular tetrahedron with the height of \( 3\sqrt6 \,\mathrm{cm} \). Find the edge length of the tetrahedron (see the picture).
\( 9\,\mathrm{cm} \)
\( 9\sqrt{2}\,\mathrm{cm} \)
\( 6\sqrt{2}\,\mathrm{cm} \)
\( 3\sqrt{6}\,\mathrm{cm} \)
2010015802
Level:
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 8\,\mathrm{cm} \). Find the distance between the point \( V \) and the line \( BD \) (see the picture).
\( 2\sqrt{17}\,\mathrm{cm} \)
\( 4\sqrt{3}\,\mathrm{cm} \)
\( 2\sqrt{19}\,\mathrm{cm} \)
\( 2\sqrt{20}\,\mathrm{cm} \)